How can we use exponents to understand fucntions?
Students will explore exponential functions by multiplying and dividing expressions with exponents. They will also learn how to graph these functions through exponential growth and exponential decay while applying to reallife functions.
Print VersionProperties of integer exponents
(N.RN.2, A.SSE.2, A.SSE.3c)
Rational exponents, applying properties of integer exponents (NRN.1, NRN.2, NRN.3)
Define, evaluate and compare exponential functions; identify domain, range and intercepts (F.IF.1, F.IF.5,
F.IF.7e)
• Classify exponential growth and decay. (F.IF.8b, F.LE.1c)
• Create and graph a function modeling two quantities. (F.IF.4, F.LE.2, F.BF.1)
Geometric sequences (F.IF.3, F.BF.2)
N.RN.1
I can define radical notation as a way to represent rational exponents.
I can explain the properties of operations of rational exponents as an extension of the properties of integer exponents.
I can explain how radical notation, rational exponents and properties of integer exponents relate to one another.
N.RN.2
I can rewrite a radical expression using rational exponents.
I can rewrite an expression with a rational exponent using a radical expression.
N.RN.3
I can find the sums and products of rational and irrational numbers.
I can recognize that the sum of a rational number and an irrational number is irrational.
I can recognize that the product of a nonzero rational number and an irrational number is irrational.
I can explain why rational numbers are closed under addition or multiplication.
A.SSE.2
I can identify various structures of expressions.
I can use the structure of an expression to identify ways to rewrite it.
I can classify expressions by structure and develop strategies to assist in classification.
A.SSE.3c
I can define an exponential function .
I can rewrite exponential functions using the properties of exponents.
I can rewrite exponential expressions to identify properties of quantities represented by the original expression.
I can explain the properties of the quantities represented by the transformed exponential expression.
F.IF.1
I can identify the domain and range of a function.
I can determine if a relation is a function.
I can determine the value of the function with proper notation (i.e. f(x)=y, the y value is the value of a function at a particular value of x.)
I can evaluate functions for given values of x.
F.IF.3
I can recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0)=f(1)=1, f(n+1)=f(n)+f(n1) for n> 1.
F.IF.4
I can define and recognize the key features in tables and graphs of linear and exponential functions.
I can identify whether the function is linear or exponential, given its table or graph.
I can interpret key features of graphs and tables of functions.
I can create a graph that models the description and indicates all of the key features of the function.
F.IF.5
I can identify and describe the domain of a function when given a graph or description of the function.
I can identify an appropriate domain based on the unit, quantity, and type of function it describes.
I can relate the domain of the function to its graph.
I can explain why a domain is appropriate for a given situation.
F.IF.7e
I can graph exponential functions, by hand in simple cases.
I can graph more complicated exponential functions using technology, showing intercepts and end behavior.
F.IF.8b
I can distinguish between exponential functions that model exponential growth and exponential decay.
I can use the properties of exponents to evaluate expressions for exponential functions in a realworld
F.LE.1c
I can identify situations that display equal ratios of change over equal intervals and can be modeled by exponential functions.
F.LE.2
I can recognize that geometric sequences can be expressed as exponential functions.
I can construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).
F.BF.1
I can define “explicit function” and “recursive process”.
I can identify the relationship between quantities in a realworld problem.
I can apply algebraic transformations of parent functions.
I can write a function that describes a realworld problem.
F.BF.2
I can identify geometric patterns in given sequences.
I can generate geometric sequences from recursive and explicit formulas.
I can use given and constructed geometric sequences to model reallife situations.
I can determine the recursive rule given geometric sequences.
I can justify the translation between the recursive form and explicit formula for geometric sequences.
I can write an explicit formula for a geometric sequence.
Print Version
exponential fucntion, exponential growth and decay, percent of increase and decrease, product of powers property, power of a power property, and power of a product property
Print VersionComing Soon.
Print Version
Thinking Strategies for Readers
Researchers who have studied the thinking processes of proficient readers conclude that if teachers taught the following strategies instead of much of the traditional skillsbased reading curriculum, students who use the strategies would be better equipped to deal with a variety of texts independently (Keene and Zimmerman, 1997). These strategies are useful for composing meaning at both a text and word level.
Monitoring for Meaning
at a text level, readers . . .
■ pause to reflect on their growing understandings
■ recognize when they understand the text, and when they don’t
■ identify when and why the meaning of the text is unclear
■ identify the ways in which a text becomes gradually more understandable by reading past an unclear portion and by rereading text
■ decide if clarifying a particular confusion is critical to overall understanding
■ explore a variety of means to remedy confusion
■ consider, and sometimes adjust, their purpose for reading
■ check, evaluate and make revisions to their evolving interpretation(s) of text
at a word level, readers . . .
■ identify confusing words
■ employ a range of options for reestablishing meaningful reading (e.g., rereading, reading on, using words around the unknown word, using letters and sounds, using a meaningful substitution)
Activating, Utilizing and Building Background Knowledge (Schema)
at a text level, readers . . .
■ activate relevant, prior knowledge before, during and after reading
■ build knowledge by deliberately assimilating new learning with their related prior knowledge
■ clarify new learning by deleting inaccurate schema
■ relate texts to their world knowledge, to other texts and to their personal experiences
■ activate their knowledge of authors, genre, and text structure to enhance understanding
■ recognize when prior knowledge is inadequate and take steps to build knowledge necessary to understand
at a word level, readers . . .
■ apply what they know about soundsletter relationships and word parts to make sense of unknown words
Asking Questions
at a textlevel, readers . . .
■ generate questions before, during and after reading about the text’s content, structure and language
■ ask questions for different purposes, including clarifying their own developing understandings, making predictions, and wondering about the choices the author made when composing
■ realize that one question may lead to others
■ pursue answers to questions
■ consider rhetorical questions inspired by the text
■ distinguish between questions that lead to essential/deeper understandings and “just curious” types of questions
■ allow selfgenerated questions to propel them through text
■ contemplate questions posed by others as inspiration for new questions
at a word level, readers . . .
■ pose selfmonitoring questions to help them understand unknown/unfamiliar words (e.g., “What would make good sense?”, “What would sound like language?”, “What would sound right and match the letters?”, “Is this a word I want to use as a writer? If so, how am I going to remember it?”)
Drawing Inferences
at a text level, readers . . .
■ draw conclusions about their reading by connecting the text with their schema
■ make, confirm, and/or revise reasonable predictions
■ know when and how to infer answers to unanswered questions
■ form unique interpretations to deepen and personalize reading experiences
■ extend their comprehension beyond literal understandings of the printed page
■ make judgments and create generalizations about what they read
■ create a sense of expectation as they read
at the word level, readers . . .
■ use context clues and their knowledge of language to predict the pronunciation and meaning of unknown/unfamiliar words
Determining Importance
at a text level, readers . . .
■ identify key ideas, themes and elements as they read
■ distinguish between important and unimportant information using their own purpose(s), as well as the text structures and word cues the author provides
■ use text structures and text features to help decide what is essential and what is extraneous
■ use their knowledge of important and relevant parts of text to prioritize what they commit to longterm memory and what they retell and/or summarize for others
■ consider the author’s bias/point of view
■ use the filter of essential/other to clarify usefulness when applying other cognitive strategies to their reading
at a word level, readers . . .
■ determine which words are essential to the meaning of the text
■ know when choosing to skip words/phrases of text will or will not impact their overall understanding
■ make decisions about when unknown/unclear words need clarification immediately and accurately, and when substitutions can be used to maintain meaning and fluency
Creating Sensory Images
at a text level, readers . . .
■ immerse themselves in rich detail as they read
■ create images connected to the senses of sight, hearing, taste, touch and smell to enhance and personalize understandings
■ attend to “heart” images – feelings evoked while reading
■ revise their images to incorporate new information and new ideas revealed in the text
■ adapt their images in response to the images shared by other readers
at a word level, readers . . .
■ use visual, auditory and kinesthetic modes when learning how words work
■ use what they know about a word’s appearance (e.g., length, spacing above and below the line) to understand unknown words
■ ask themselves “Does that look right?” and “Does that sound right?” whencrosschecking unknown words
Synthesizing Information
at a text level, readers . . .
■ continually monitor overall meaning, important concepts and themes while reading
■ recognize ways in which text elements fit together to create larger meaning
■ create new and personal meaning
■ develop holistic and/or thematic statements which encapsulate the overall meaning of the text
■ capitalize on opportunities to share, recommend and criticize books
■ attend to the evolution of their thoughts across time while reading a text, and while reading many texts
at a word level, readers . . .
■ select specific vocabulary from the text(s) to include in their synthesis because they know that specific language is highly meaningladen
■ know when certain vocabulary is critical to the text’s overall meaning, and therefore, must be understood if comprehension is to be achieved
Problem Solving
at a text level, readers . . .
■ know that once meaning has broken down, that any of the other cognitive behaviors can be employed to repair understanding
■ use information from the three deep surface structure systems to repair text meaning
at a word level, readers . . .
■ use information from the three surface structure systems to solve word issues
■ select from a wide range of word strategies (e.g., skip and read on, reread, use context clues, use the letters and sounds, speak to a peer reader) to help make sense of unknown words
■ develop reading fluency
Print VersionThoughtful Strategies by Learning Style
Mastery  Interpersonal  Understanding  Self – Expressive  Utility (Can be used in multiple styles) 
Fact or Fiction Spider/Fist List Word Association Word Wall Reading for Meaning Interactive Lecture Group & Labeling Categories Memory Box Write to Learn Building Writing  Reciprocal Learning Think/Pair/Share Give one, Get one Collaborative Summarizing Jeopardy  Anticipation Guides KWL Concept Attainment Compare/Contrast 1,2,3,4 Yes, No, Why  EtchaSketch Mystery  Graduated Difficulty Comprehension Menu Task Rotation Voc Notebook Carousel Brainstorming Boggle Reader’s Theatre Vocabulary Code TGT Jigsaw 421 Free Write Kindling

The biggies…. The following strategies require a little more planning to use. They are all very effective. Your school has folders and materials that specifically explain these strategies.
Word WorksCracking Vocabulary’s CODE – 4 phases of vocabulary learning
 Connect – new words to prior knowledge
 Organize – new words to find relationships
 Deepprocessing – internalizing the words, deepen the understanding
 Exercise – or rehearse their knowledge of new words
Reading for Meaning – strategy that helps students become proficient at making claims, finding main ideas, and using reasoning and details to support their ideas.
· Students are presented with a series of statements before they read the text, they need to either agree/disagree with the statement.
· After reviewing the statements, the students read the text and collect evidence either for or against the statements.
Task Rotation – creating activities that fit students’ learning styles:
· Mastery
· Interpersonal
· Understanding
· Selfexpressive
Interactive Lecture – strategy that increases the students’ ability to remember, comprehend, and think actively about a lectures’ content. It engages the students by moving them through the four phases:
 Connect – hook students’ attention
 Organize – use graphic organizer/note taking procedure to help organize info
 Deepprocessing – pause every 57 min during lecture to allow students time to process information with questions in the different learning styles
 Exercise – use review questions and have students use notes in a synthesis or application task.
Activities/Tools
These activities or tools are easy to slip in anywhere within your unit plan. They can be used for the CODE strategies, class openers, to brainstorm, to review, or as energizers for those “glazed over” moments. They are categorized somewhat, but several of these activities can be used in more than one category. Many of the strategies are referenced (in parenthesis) if you want more information on each of these activities.
Class Openers
Fact or Fiction/ Three’s a Crowd – Students decide which word/fact of three doesn’t belong and explain why. (Tool book p.10)
Anticipation Guides – Teacher prepares 35 statements based on the content that the students will be reading. Students are asked to decide which statements they believe the text will support. Teacher develops a class tally for each statement and discusses opinions. Students then read text. (Tool book p. 40)
Give one, Get one students generate ideas from a question posed by the teacher, then have to collect a predetermined number of ideas from their classmates. (Tool book, p.11)
KWL Tool to assess students’ prior knowledge, help generate questions about what they want to learn, and encourage reflection about what they have learned. (Tool book, p.28)
Spider List/Fist list/Fishbone teacher provides a category in palm of hand/belly of spider and the students brainstorm ideas to fill in the fingers/legs/bones of hand print, spider, or fish sketch or vice versa.
Word Association/3way tie students select 3 words from a unit vocabulary and arrange them in a triangle. They then connect the words with lines then write a sentence that describes the relationship between the words that are connected. (binder)
Content Teaching
Word Wall – Collection of words on the wall for students to use during their reading and writing (Binder)
Reciprocal Learning/Peer Practice Strategy students work in pairs (player and coach) to review or read & summarize concepts.
Think/pair/shareteacher poses a question, the students think and construct a response, then share their ideas with a neighbor, teacher records/collects their ideas. (Tool book, p.10)
Vocabulary Notebook Where students collect critical vocabulary In the notebook students can write their initial “educated” definitions, then they can write the dictionary definition, and maybe a visual image as well. There is a lot of variations to this one. (Tool book, p. 92; binder)
Group & Labeling – students examine a list of vocabulary words and place them into groups based on common characteristics. For each group that students create, they devise a label that describes
Etchasketch – students draw pictures, symbols, or icons to represent ideas presented in a lecture, reading, or other form of presentation (Tool book, p. 60)
Collaborative Summarizing – After lecture or reading, the students are asked to identify the 36 most important ideas. Students then pair up and compare their lists and come up with a consensus of the most important ideas with their partner. They (the group of two) pair up with another group of two and compare lists and once again come with an agreed upon list of 36 important ideas. These four use their list to create a collaborative summary. (Tool book, p.78)
Jigsaw students work cooperatively with each student having an assigned task within the group to accomplish/perform.
Carousel Brainstorming – teacher generates different styles of questions & posts them around the room. The students work in groups of 35, rotates around the room to reading the question, the other responses, and either expands on existing ideas or develops a new idea. (Tool book, p.19)
421 Free Write – students identify 4 important ideas previously presented in the lesson. Each student meets with another student to compare ideas and decide on the two most important from their lists of four. This pair meets with another pair. They compare their ideas, then come to a consensus on the most important idea. The students then take this and do a free write, explaining all they know about the big idea. (Tool book, p. 82)
Concept Attainment – teacher presents examples and nonexamples of a concept in a class discussion, the students use these to brainstorm the key attributes/characteristics of the concept.
Compare/Contrast – comparing likenesses and differences. The Georgia website I sent you has several different variations on this that are interesting.
Kindling – F  Find a question that can be explored
I  Internalize the question
R  Record your thoughts (sketch, write…)
E  Exchange ideas with a neighbor
S  Select and record your ideas in public (Tool Book p.74)
Comprehension Menu – an abbreviated version of Task Rotation. Teacher creates at least four questions in the four learning styles about the content. (Tool book, 162)
Review Activities/Tools
1,2,3,4 – teacher stops 5 minutes before end of class period to allow the students to reflect upon what was presented by writing in this format:
1 – What was the big idea
2 – Important details discussed
3 – Personal connections discovered
4 – Questions students have about the content
Boggle – students review notes for 2 minutes, then list as many ideas or details they can remember for 25 minutes, then students share their ideas with 1 or 2 other students and can add to their lists. Lastly, students pair up and compete with another student using the Boggle technique (They earn a point for every idea that their Boggle partner doesn’t have). Then the students go back to their study teams and compute the team score. (Tool book p.134)
Memory Box – a box usually put on a test where students can take the first five minutes to list as many things they can remember(formulas, definitions, etc…)
TGT – (Teams Game Tournament) teacher creates vocabulary/fact cards. The teacher divides the students heterogeneously by academic ability. This is the study group. After they have studied for a while, they then move to homogenous groups established by the teacher and compete against each other. They follow the points system to see how many points they take back to their study teams. This is a great review activity, it takes some time to prepare it, but it is well worth it.
Jeopardy  This follows the same format as the game show. It is great to use on the active board. There are many already developed on the Ashland Schools website.
Categories – technique for forming groups and reviewing content. (Tool book,
p. 138)
Tool Book refers to: Silver & Strong, 2001 Tools for Promoting Active, InDepth Learning, Thoughtful Education Press.
Print Version